Instances of 'Texians' in newspaper articles published before October 13, 1835
With regard to the earliest print occurrence of Texian, I note that an Elephind newspaper database search turns up four occurrences of the plural form Texians prior to October 13, 1835—the date when, according to the community wiki answer, Texian appeared in print in the New Orleans Bee.
The first instance is from a newspaper in Brazoria, Texas; the next two from Leesburg, Virginia, and Richmond, Virginia (the first citing a newspaper in Opelousas, Louisiana, as its source and the second citing the New Orleans Bee); and the fourth from the same Texas newspaper that later reprinted the Bee's article of October 13.
Here are the four instances.
From a speech by Col. Stephen F. Austin to the Committee of Brazoria, reported in the [Brazoria, Texas] Texas Republican (September 19, 1835):
If any acts of imprudence have been committed by individuals they certainly resulted from the revolutionary state of the whole nation [of Mexico], the imprudent and censurable conduct of the State authorities and the totl want of a local Government in Texas. It is indeed a source of surprise and credible congratulation that so few acts of this description have occurred under the peculiar circumstances of the times. It is however to be remembered that acts of this nature were not the acts of the people, nor is Texas responsible for them. They were as I before observed the natural consequence of the revolutionary state of the Mexican Nation and Texas certainly did not originate that revolution, neither have the people, as a people, participated in it. The consciences and the hands of the Texians are free from censure, and clean.
This country [Mexico] is now in anarchy, threatened with hostilities, armed vessels are capturing every thing they can catch on the coast, and acts of piracy are said to be committed under the cover of the Mexican flag. Can this state of things exist without precipitating the country into a war? I think it c[a]nnot, and therefore believe that it is our bounden and solemn duty, as Mexicans and Texians to represent the evils that are likely to result from this mistaken and most impolitic policy in the military movements.
From an untitled item in the [Leesburg, Virginia] Genius of Liberty (September 26, 1835), citing an original story in the Opelousas [Louisiana] Gazette of unspecified earlier date:
The Opelousas (Lou.) Gazette, contemplating the probability of a war between the Texians and Santa Anna, thus compassionately and respectfully speaks of the belligerents:—
"You are welcome to the combat gentlemen. One goos result is certain—the world will lose many bad citizens, and the devil will gain some faithful servants.
"Every body knows that Texas has been to the United States what Botany Bay has been to Great Britain. The emigrants thither, like the followers of king David in the cave of Adullum, have been all those who were oppressed, and all those who were in debt—in other words, vagabonds and refugees from justice. To read the thundering manifestoes of these fellows, who know nothing about republics, and care nothing about liberty, one would think they had been hired to bring the immortal doctrines of Plato into ridicule and contempt."
From an untitled item in the Richmond [Virginia] Enquirer (October 9, 1835), reprinted from an item in the New Orleans Bee of unspecified date:
A. Prize.—The notorious Thompson has been lakei it Inst. The schooner San Felipe had sailed from this port on the 25th of last month [presumably August 1835] for Brassoria, with Colonel Stephen Austin and other Texians, and on the evening of Tuesday following, she heard firing ahead. On nearing the place. Capt. Hurd perceived the Mexican schooner Correo, commanded by Thompson, and an armed sloop engaged in attacking the American brig Tremont, which was assisted by a steamboat. When Captain Hurd advanced, the Mexican ships ceased and retreated; and then the steamboat took much of the cargo and most of the passengers on board, and brought them into Brassoria.
The fourth instance, from in the [San Felipe de Austin, Texas] Telegraph and Texas Register (October 10, 1835), is simply a reprint of Austin's speech of September 8, 1835, in Brazoria, first reported in the September 19, 1835, Texas Republican, as noted above.
So we have two instances of Texians from September 8, 1835, in a speech by Stephen F. Austin. Then we have an instance from a newspaper in Opelousas, Louisiana, at some date before September 26, 1835. And then we have an instance from the New Orleans Bee at some date before October 9, 1835.
It us certainly not impossible that the instance from the Opelousas Gazette was published before Austin's speech—but the facts that Austin departed from New Orleans on August 25 and used the term Texians on his arrival in Brazoria on September 8 strongly suggest that Austin was familiar with the term prior to his return to Texas.
It isn't entirely clear whether Austin learned the term while in Louisiana, picked it up while in prison in Mexico City, or remembered it from his earlier years in Texas. He had emigrated to the Texas in 1825, leaving it in 1833 as a delegate for colonists of U.S. origin who were seeking separate federal Mexican statehood for Texas, was imprisoned in Mexico City, for his advocacy in 1834, and released in a general amnesty in July 1835. According to the Wikipedia article about Austin, "in August 1835 [he] left Mexico to return to Texas via New Orleans." However, the timeline indicates that his stay in New Orleans was rather brief, and he seems to have expected his hearers in Brazoria to be familiar with the word Texians, so I'm inclined to think that it was current among the colonists in his area by 1833.
The earliest Elephind match for 'Texans' in a newspaper story
By contrast, the first instance of Texans that an Elephind newspaper search finds is from an untitled article in the [San Felipe de Austin, Texas] Telegraph and Texas Register of November 7, 1835—the article reprinted from the New Orleans Bee of October 13, 1835 (and cited in the community wiki answer to this question) that assesses the merits of Texians versus Texans, Texonians, Texasians, Texicans, and Texasites as a designation for "the people of Texas."
Early instances of 'Texonians' in Elephind search results
Although the New Orleans Bee asserts in its October 13, 1835 article that "Texonian and Texasite are absurd epithets," Texonians is by eight full months the earliest of the various options it lists (including Texan and Texian) to appear in Elephind search results.
From an untitled item and reprinted in the [Lawrenceburg, Indiana] Indiana Palladium (January 10, 1835), reprinted from the [New Orleans] Louisiana Advertiser of unspecified date:
He [Mr. Butler, the U.S. minister to Mexico] has no objection to their [the British] investing a port on the southern extremity of Mexico [as a naval depot for the British navy], and concludes by pledging the honor of the nation for their [the Americans living in the area of Galveston Bay proposed for the new depot] support and protection, in case the British government were inclined to proceed to coercive measures. The Mexican government had, by the last advices, made no reply; but their partiality for the English, and their hatred to the Texonians, connected with their intestine feuds and the depreciated state of their resources, would, it is generally believed, act as an inducement to their acquiescence.
Three other early instances of Texonians—two from August 1835 and one from October 2, 1835—are worthy of note.
From an untitled item in the Richmond [Virginia] Enquirer (August 7, 1835), reprinted from the [New Orleans] American (July 17, 1835):
It is impossible for Texas to remain long under the dominion of Mexico. The character of the Texonians who are generally emigrants from the U. States, is too essentially different from that of the Mexicans, for them to remain long attached to the uncongenial laws and customs of Mexico. The Texonians are too far ahead of their present would-be-masters,—they know too much of the principles of republicanism, and are too much attached to the free institutions they have been taught from childhood to appreciate and revere, to allow themselves to be trampled upon.
From an untitled item in the Crawfordsville [Indiana] Record (August 22, 1835), reprinted from the Mobile [Alabama] Register (July 20, 1835):
Notwithstanding these conflicting opinions as to Santa Anna's objects, we have reason to believe that actual hostilities have been commenced by a portion of the Americans in Texas. A paragraph in the New Orleans Bulletin of Saturday says, that, captain Moore, of the schooner Shenandoah, in a short run from Brazoria, reports that the Texonians had seized upon the fort at Annahuac, garrisoned by one hundred men, whom they captured and sent to St. Felipe, A portion of the people thought this step was pushing matters to extremities, and one was uncalled for, but all were determined to maintain their just rights should they be assailed by the Mexican powers."
And from "Interesting from Texas," in the Richmond [Virginia] Enquirer (October 2, 1835), reprinted from the New Orleans True American, via the Philadelphia Inquirer:
The Convention was to have met on the 14th September, and it was expected that it would take such measures as will excite Santa Anna to prosecute his threatened invasion. It will doubtless call upon every Texonian to resist, by every honorable means—remonstrances first, and arms afterwards—the usurpation of Centralism.
The Texonians look with confidence towards their fellow-citizens of the United States, particularly to those of the Western States, for assistance in case of a war with Santa Anna. It is hoped that they will not be disappointed in this expectation.
As all four these instances of Texonians come directly or indirectly from New Orleans newspapers, it seems highly likely that Texonian was the dominant term for "Anglo-American resident of Texas" in New Orleans in 1835, until Texian emerged as an alternative—in all likelihood through its use by contemporaneous residents of Texas such as Stephen F. Austin.
If not for their influence at a critical early time, we might today be referring to the denizens of the Lone Star State as Texonians—and perhaps (by back formation) referring to the state itself as Texford.